Question

Which of the following points shows a solution to the simultaneous inequalities y>12x+5
𝑦
>
1
2
𝑥
+
5
and y<−2x+1
𝑦
<
−
2
𝑥
+
1
?(1 point)
Responses

(1,5.5)
(
1
,
5.5
)
left parenthesis 1 comma 5.5 right parenthesis

(4,8)
(
4
,
8
)
left parenthesis 4 comma 8 right parenthesis

(−1.6,4.2)
(
−
1.6
,
4.2
)
left parenthesis negative 1.6 comma 4.2 right parenthesis

(−5,5)

Answer 1

To determine which point satisfies the simultaneous inequalities \(y > 12x + 5\), \(y > \frac{1}{2}x + 5\), and \(y < -2x + 1\), we can evaluate each point one by one.

1. **Point (1, 5.5):**
- Check \(y > 12x + 5\):
\[
5.5 > 12(1) + 5 \rightarrow 5.5 > 17 \quad \text{(False)}
\]
- Check \(y > \frac{1}{2}x + 5\):
\[
5.5 > \frac{1}{2}(1) + 5 \rightarrow 5.5 > 5.5 \quad \text{(False, not strictly greater)}
\]
- Check \(y < -2x + 1\):
\[
5.5 < -2(1) + 1 \rightarrow 5.5 < -1 \quad \text{(False)}
\]
- **Point (1, 5.5) does not satisfy all inequalities.**

2. **Point (4, 8):**
- Check \(y > 12x + 5\):
\[
8 > 12(4) + 5 \rightarrow 8 > 53 \quad \text{(False)}
\]
- Check \(y > \frac{1}{2}x + 5\):
\[
8 > \frac{1}{2}(4) + 5 \rightarrow 8 > 7 \quad \text{(True)}
\]
- Check \(y < -2x + 1\):
\[
8 < -2(4) + 1 \rightarrow 8 < -7 \quad \text{(False)}
\]
- **Point (4, 8) does not satisfy all inequalities.**

3. **Point (-1.6, 4.2):**
- Check \(y > 12x + 5\):
\[
4.2 > 12(-1.6) + 5 \rightarrow 4.2 > -19.2 + 5 \rightarrow 4.2 > -14.2 \quad \text{(True)}
\]
- Check \(y > \frac{1}{2}x + 5\):
\[
4.2 > \frac{1}{2}(-1.6) + 5 \rightarrow 4.2 > -0.8 + 5 \rightarrow 4.2 > 4.2 \quad \text{(False, not strictly greater)}
\]
- Check \(y < -2x + 1\):
\[
4.2 < -2(-1.6) + 1 \rightarrow 4.2 < 3.2 + 1 \rightarrow 4.2 < 4.2 \quad \text{(False, not strictly less)}
\]
- **Point (-1.6, 4.2) does not satisfy all inequalities.**

4. **Point (-5, 5):**
- Check \(y > 12x + 5\):
\[
5 > 12(-5) + 5 \rightarrow 5 > -60 + 5 \rightarrow 5 > -55 \quad \text{(True)}
\]
- Check \(y > \frac{1}{2}x + 5\):
\[
5 > \frac{1}{2}(-5) + 5 \rightarrow 5 > -2.5 + 5 \rightarrow 5 > 2.5 \quad \text{(True)}
\]
- Check \(y < -2x + 1\):
\[
5 < -2(-5) + 1 \rightarrow 5 < 10 + 1 \rightarrow 5 < 11 \quad \text{(True)}
\]
- **Point (-5, 5) satisfies all inequalities.**

Therefore, the correct point that shows a solution to the simultaneous inequalities is \((-5, 5)\).